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A right-flat tensor bimodule can have nonprojective output

Example

Let $k$ be a field, let $A = k$ with its trivial grading and let $B = k [ε] / (ε^{2})$ with $ε$ placed in degree $0$ , so that $B$ is a graded $k$ -algebra concentrated in degree $0$ . Let $π : B ↠ k$ be the augmentation with $π (ε) = 0$ , and let $M = k$ be the graded $(B, A)$ -bimodule concentrated in degree $0$ whose left $B$ -action is $b \cdot m : = π (b) m$ and whose right $k$ -action is ordinary multiplication.

Then $M$ is flat as a right $k$ -module, so $M \otimes_{k} -$ is exact, but the left $B$ -module $M \otimes_{k} k ≅ k$ is not projective. Thus right $A$ -flatness of the bimodule does not imply that its tensor functor carries finite graded projectives to projective outputs.

Facts & Assumptions

Given: A field $k$ , the graded $k$ -algebras $A = k$ and $B = k [ε] / (ε^{2})$ in degree $0$ , the augmentation $π : B \to k$ , and the graded $(B, A)$ -bimodule $M = k$ with $b \cdot m = π (b) m$ and $m \cdot λ = m λ$ .

[L1]

Graded algebras, graded modules and degree-zero maps are defined in Associative graded algebras, bimodules, and internal shifts; since every module here is concentrated in degree $0$ , all module maps are degree-zero.

[L2]

The tensor product of graded modules carries the total-degree grading and the outer action $b (m \otimes n) = (b m) \otimes n$ (Graded balanced tensor product and homogeneous Hom).

[L3]

If $M$ is flat as a right $A$ -module then $M \otimes_{A} -$ is exact, and if $M$ is finite graded projective as a left $B$ -module then $M \otimes_{A} -$ preserves finite graded projectives (Bimodule tensor exactness and preservation of finite projectives have separate hypotheses).

[L4]

Projective objects have the lifting property, and a finite direct sum of shifts $B {s_{1}} \oplus \dots \oplus B {s_{n}}$ is finite graded projective (Finite graded projectives are finite shifted-free summands).

Verification

1.1

The two actions on $M$ commute: $(b \cdot m) \cdot λ = π (b) m λ = b \cdot (m λ)$ , and each is additive and unital, so $M$ is a $(B, A)$ -bimodule; both actions preserve the degree- $0$ part because $π$ and the scalar action do, so $M$ is a graded bimodule.

1.2

$M = k$ is a free right $k$ -module of rank one, hence flat, so $M \otimes_{k} -$ is exact; equivalently the functor is $k \otimes_{k} -$ , which is naturally the identity on $k$ -vector spaces.

1.3

The unit isomorphism $M \otimes_{A} A \to M$ , $m \otimes λ \mapsto m λ$ , identifies $M \otimes_{k} k$ with $k$ , and under this identification the left $B$ -action is $b \cdot (m \otimes λ) = b m \otimes λ$ , i.e. the action of $B$ on $k$ through $π$ ; so $M \otimes_{k} k ≅ k$ as graded left $B$ -modules.

1.4

The module $k = B / (ε)$ is not projective as a left $B$ -module. The quotient map $π : B ↠ k = B / (ε)$ is $B$ -linear and degree-zero; if $k$ were projective, its lifting property against $π$ and the identity of $k$ would produce a $B$ -linear section $s : k \to B$ with $π s = 1_{k}$ . Writing $u : = s (1)$ one has $π (u) = 1$ , so $u = 1 + c ε$ for some $c \in k$ , and $B$ -linearity gives $ε u = s (ε \cdot 1) = s (0) = 0$ , whereas $ε u = ε + c ε^{2} = ε \neq 0$ . This contradiction shows that no such section exists, so $k$ is not projective over $B$ .

L1L3L4

2.1

Steps 1.2 and 1.3 give a right-flat bimodule $M$ whose tensor functor is exact and whose value on the finite graded projective left $A$ -module $A = k$ is $M \otimes_{k} k ≅ k$ ; by step 1.4 that output is not projective as a left $B$ -module, and it is not a finite graded projective module either. Hence the exactness hypothesis of [L3] does not deliver its projectivity conclusion, which is why that conclusion carries the separate hypothesis that $M$ be finite graded projective over $B$ — a hypothesis $M$ fails by step 1.4.

step 1.2step 1.3step 1.4L3L4

3.1

The example therefore exhibits a right-flat tensor bimodule whose tensor functor is exact but which produces a nonprojective, non-finite-projective output from a finite graded projective input. ∎

step 2.1

Depends on

Used by

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Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §§2a-2b, author pp. 8-9 (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, ch. 3, §3.2, printed pp. 68-69 (standard reference, not scraped)